Step 2: The controller

First, make the controller board (instructions here), drill the holes and tin the pads carefully. Place all the components as shown in the picture and solder them in place.

Step 3: The cube

Mark a 3x3 1 inch grid in a corck and hole the places for the LEDs, this will keep them aligned while you bend and solder the leads. Place them diagonally so the leads won't touch, solder the cathodes to one side and then bend the anodes above the cathodes. Solder the anodes 90º away from the cathodes.

Repeat the process to make two more grids and place them on each other making anodes and cathodes be in the same side. Bend the 9 protuding cathodes down and solder them in 3 columns.

I've bent the 9 anodes so they came arranged in three groups of three, you can simply solder a ribbon cable, but it won't hold the cube up as a structure.